The twisted Hecke algebras

• The affine Weyl group for two characters
• The Hecke algebra

The affine Weyl group for two characters

For $\chi, \chi^\prime \colon T(k_F) \to \mathbb{C}^\times$, we looked at this bi-torsor $$ \tilde{W}_ \chi \curvearrowright {}_ \chi \tilde{W}_ {\chi^\prime} \curvearrowleft \tilde{W}_ {\chi^\prime}. $$ Inside $\tilde{W}_ \chi$ we had this subgroup $$ \tilde{W}_ \chi \supseteq \tilde{W}_ \chi^0 = \langle s_\alpha : \alpha \in \Phi_{\chi,\mathrm{aff}} \rangle. $$ Then we have $$ 1 \to \tilde{W}_ \chi^0 \to \tilde{W}_ \chi \to \Omega_\chi \to 1, $$ where $\Omega_\chi$ is the subgroup of with $\chi$-length zero.

Now we have an isomorphism $$ \tilde{W}_ \chi^0 \backslash {}_ \chi \tilde{W}_ {\chi^\prime} \cong {}_ \chi \tilde{W}_ {\chi^\prime} / \tilde{W}_ {\chi^\prime}^0 $$ and so we get an isomorphism $$ \Omega_\chi \cong {}_ \chi \Omega_{\chi^\prime} \cong \Omega_{\chi^\prime} $$ where we have $$ {}_ \chi \Omega_{\chi^\prime} \cong \lbrace w \in {}_ \chi \tilde{W}_ {\chi^\prime} : w(\Phi_{\chi^\prime,\mathrm{aff}}^+) = \Phi_{\chi,\mathrm{aff}}^+ \rbrace. $$

Lemma 1. 1. Let $\omega \in {}_ \chi \Omega_{\chi^\prime}$. Then $\ell(w) \lt \ell(wv^\prime)$ for any $v^\prime \in \tilde{W}_ {\chi^\prime}^0 - \lbrace 1 \rbrace$.

1. Let $v_1 \le_{\chi^\prime} v_w$ in the Bruhat order of $\tilde{W}_ {\chi^\prime}^0$. Then $w v_1 \le w v_2$ in the Bruhat order of $\tilde{W}$.

Definition 2. Let $w \in {}_ \chi \Omega_{\chi^\prime}$. For $v \in w \tilde{W}_ {\chi^\prime}^0 = \tilde{W}_ \chi^0 w$ we define $$ ell_w(v) = \ell_{\chi^\prime}(w^{-1} v) = \ell_\chi(v w^{-1}). $$ We also say $v \le_w v^\prime$ if $w^{-1} v \le_{\chi^\prime} w^{-1} v^\prime$.

We can also think about this as $$ \ell_w(v) = \hash \lbrace \alpha \in \Phi_{\alpha,\mathrm{aff}}^+ : v(\alpha) \lt 0 \rbrace. $$

Lemma 3. If we write $$ v = s_{i_1} \dotsc s_{i_n} \tau $$ where $\tau \in \Omega$ and $s_{i_j}$ are the simple reflections in $\tilde{W}$, then $$ \ell_w(v) = \hash \lbrace j : s_{i_j} \in \tilde{W}_ {\chi_j}^0 \rbrace, $$ where $\chi_j = s_{i_{j+1}} \dotsm s_{i_n} \tau \chi^\prime$.

The Hecke algebra

Recall we had $$ {}_ \chi \mathcal{H}_ {\chi^\prime} = \bigoplus_{w \in \tilde{W}} {}_ \chi \mathcal{H}_ {\chi^\prime}^w $$ where we have ${}_ \chi \mathcal{H}_ {\chi^\prime}^w \neq 0$ if and only if $w \in {}_ \chi \tilde{W}_ {\chi^\prime}$. This was $$ {}_ \chi \mathcal{H}_ {\chi^\prime}^w = \lbrace f \colon IwI \to \mathbb{C} : f(k_1 g k_2) = \chi(k_1) f(g) \chi^\prime(k_2) \rbrace. $$ These are one-dimensional (unless it is zero) but doesn’t have a canonical generator. For weach $w \in {}_ \chi \tilde{W}_ {\chi^\prime}$ we choose a lift $\dot{w} \in N_G(T)(F)$ and set $$ {}_ \chi T_{\chi^\prime}^w(\dot{w}) = 1. $$ Then we have for $w \in {}_ \chi \tilde{W}_ {\chi^\prime}$ and $v \in {}_ {\chi^\prime} \tilde{W}_ {\chi^{\prime\prime}}$ the formula $$ ({}_ \chi T_{\chi^\prime}^{\dot{w}} {}_ {\chi^\prime} T_{\chi^{\prime\prime}}^{\dot{v}})(g) = \sum_{g^\prime \in Iv^{-1}I/I} {}_ \chi T_{\chi^\prime}^{\dot{w}}(g g^\prime) {}_ {\chi^\prime} T_{\chi^{\prime\prime}}^{\dot{v}}(g^{\prime-1}) = \sum_{g^{\prime\prime} \in I^+/I^+ \cap v^{-1} I^+ v} {}_ \chi T_{\chi^\prime}^{\dot{w}}(g g^{\prime\prime} \dot{v}^{-1}). $$

Lemma 4. If $\ell(wv) = \ell(w) + \ell(v)$ then we have $$ {}_ \chi T_{\chi^\prime}^{\dot{w}} {}_ {\chi^\prime} T_{\chi^{\prime\prime}}^{\dot{v}} = {}_ \chi T_{\chi^{\prime\prime}}^{\dot{w} \dot{v}}. $$ For $s$ a simple reflection in $\tilde{W}$, we have $$ {}_ \chi T_{s\chi}^{\dot{s}} {}_ {s\chi} T_\chi^{\dot{s}} = q \chi(\dot{s}^{-2}) {}_ \chi T_\chi^1 + (q-1) 1_{\tilde{W}_ \chi^0}(s) {}_ \chi T_\chi^{\dot{s}}. $$

Example 5. For $G = \mathrm{SL}_ 2(\mathbb{Q}_ p)$ and the nontrivial character $\chi \colon k_F^\times \to \lbrace \pm 1 \rbrace$, we have for $\dot{s} = (\begin{smallmatrix} & 1 \br -1 \end{smallmatrix})$ that $$ ({}_ \chi T_\chi^{\dot{s}})^2 = p \Bigl( \frac{-1}{p} \Bigr) {}_ \chi T_\chi^1. $$

Proposition 6. 1. Let $w_1 \in {}_ \chi \tilde{W}_ {\chi^\prime}$ and $w_2 \in {}_ {\chi^\prime} \tilde{W}_ {\chi^{\prime\prime}}$ with $w = w_1 w_2 \in {}_ \chi \tilde{W}_ {\chi^{\prime\prime}}$. If $$ \ell_{\omega_1}(w_1) + \ell_{\omega_2}(w_2) = \ell_{\omega}(w) $$ then we have $$ {}_ \chi T_{\chi^\prime}^{\dot{w}_ 1} {}_ {\chi^\prime} T_{\chi^{\prime\prime}}^{\dot{w}_ 2} = c {}_ \chi T_{\chi^{\prime\prime}}^{\dot{w}} $$ for some $c \neq 0$.

1. Let $\alpha \in \Phi_{\chi,\mathrm{aff}}^+$ be a simple affine root. Then we have $$ ({}_ \chi T_\chi^{\dot{s}_ \alpha})^2 = a {}_ \chi T_\chi^{\dot{s}_ \alpha} + b {}_ \chi T_\chi^1 $$ for some $a$ and $b \neq 0$.

Here note that $\alpha$ need not be a simple reflection in $\tilde{W}$. For the first identity, we first check this in the case when $w_2 = \omega_2 \in {}_ {\chi^\prime} \Omega_{\chi^{\prime\prime}}$. We induct on the length of $\omega_2$ and write $\omega_2 = \omega_2^\prime s_i$. Then we have $$ {}_ \chi T_ {\chi^\prime}^{\dot{w}_ 1} {}_ {\chi^\prime} T_ {\chi^{\prime\prime}}^{\dot{w}_ 2} = c {}_ \chi T_ {s_i \chi^{\prime\prime}}^{\dot{w}_ 1 \dot{\omega}_ 2^\prime} {}_ {s_i \chi^{\prime\prime}} T_{\chi^{\prime\prime}}^{\dot{s}_ i}. $$ Now the point is that ${s_i \chi^{\prime\prime}} T_{\chi^{\prime\prime}}^{\dot{s}_ i}$ squares to a constant, so we can multiply the right hand side up to a constant, regardless of whether the length increases or decreases.

Lemma 7. Suppose there exists a one-dimensional representation $M$ of ${}_ \chi \mathcal{H}_ \chi$. Then we can normalize the ${}_ \chi \tilde{T}_ \chi^w$ so that

1. ${}_ \chi \tilde{T}_ \chi^w {}_ \chi \tilde{T}_ \chi^v = {}_ \chi \tilde{W}_ \chi^{wv}$ when $\ell_{\omega_1}(w) + \ell_{\omega_2}(v) = \ell_\omega(wv)$,
2. $({}_ \chi \tilde{T}_ \chi^s)^2 = (q-1) {}_ \chi \tilde{T}_ \chi^s + q$.